Kilmer did a pretty good job but the film itself portayed Morrison as little more than the stereotypical, drug and alcohol induced rock star.
Here's what wiki says:
James Douglas "Jim" Morrison (December 8, 1943 – July 3, 1971) was an American singer-songwriter and poet, best remembered as the lead singer of Los Angeles rock band The Doors. Following The Doors' explosive rise to fame in 1967, Morrison developed a severe alcohol and drug dependency that culminated in his death at the age of 27 in Paris. He is alleged to have died of a heroin overdose, but as no autopsy was performed, the exact cause of his death is still disputed. Morrison was well known for often improvising spoken word poetry passages while the band played live. Due to his wild personality and performances, he is regarded by critics and fans as one of the most iconic, charismatic and pioneering frontmen in rock music history. Morrison was ranked number 47 on Rolling Stone's list of the "100 Greatest Singers of All Time", and number 22 on Classic Rock Magazine's "50 Greatest Singers In Rock".